Saturday, 09 April 2022 01:22

VORONOY, Georgy (1868-1908)

Georgy Voronoy's name is sometimes written as Georgii Voronoi. His father owned property at Zhuravka, Poltava province, where Georgy was born. His father was a professor at the Nezhinsk lycée, then went on to become director of the Gymnasium at Kishinev, then the Gymnasium at Berdyansk, and finally the Gymnasium at Priluki. Georgy completed his high school education at the gymnasium in Priluki, graduating in 1885. During his time at this Gymnasium he had already shown his mathematical brilliance, being particularly fond of algebra. He solved a problem posed by Professor Ermakov of Kiev University on factorising polynomials which had appeared in the Journal of Elementary Mathematics. Although still a school pupil, this led to his first publication - in fact a quite substantial article.

He then entered the University of St Petersburg, joining the Faculty of Physics and Mathematics. There he studied Serret's algebra text and Faà di Bruno's work on binary forms. He again was attracted to algebra and found Sokhotsky's course the most enjoyable of all. He wrote in his diary (see for example [2]):

- The pure mathematics lectures captivate me more and more. I prefer Professor Sokhotsky's lectures in the special course on higher algebra to all the others. ... The main thing that concerns me is whether I have enough talent.

There is no doubt that he had ample talent and his subsequent accomplishments show this very clearly, yet it says something of his personality that he confided his doubts to his diary. Voronoy did not have a particularly easy time at university. By this time his father had retired and although he had held posts of considerable standing, nevertheless he was not well off after retirement and could not fully support his son. Voronoy had to tutor mathematics, often for rather small fees, and, taking such work very seriously and putting much effort into it, he found giving lessons exhausting. He also found life in a student residence difficult and he writes frequently in his diary about the difficulty of studying with so much noise going on and frequent student disturbances.

After graduating from St Petersburg in 1889, writing a dissertation on Bernoulli numbers, Voronoy decided to remain there and work for his teaching qualification. He was awarded a Master's Degree in 1894 for a dissertation on the algebraic integers associated with the roots of an irreducible cubic equation. He wrote in the Preface to the dissertation:

- In the essay I am now presenting, results from the general theory of algebraic integers are applied to the particular case of numbers depending on the root of an irreducible equation x3 = rx + s.

The results obtained turn out to be very graphic. All integers of the field in question have a certain form. Using the form taken by the integers, it is not difficult to find a form embracing all the integral numbers divisible by a given ideal number, or, in other words, to find the ideal corresponding to that given ideal number. In our exposition the resolution of these questions is based on a detailed study of the solutions of third-degree equations relative to a prime and a composite modulus.

Voronoy lectured at Warsaw University, being appointed professor of pure mathematics there in 1894. He wrote his doctoral thesis on algorithms for continued fractions which he submitted to the University of St Petersburg. He writes in the Preface to his thesis On a generalisation of the continued fraction algorithm (Russian):

- In his article 'De relatione inter ternas pluresve quantitates instituenda' Euler gave the first generalisation of the continued fraction algorithm. ... Jacobi changed Euler's algorithm somewhat in order to unify the calculations.

Voronoy then discussed Jacobi's generalisation, after which he went on to look at generalisations due to Poincaré and Hurwitz. He then looked at contributions by Dirichlet and Hermite, showing that none of the above provided a satisfactory generalisation. He then writes:

- Taking as the basis of our investigations a special way of viewing the continued fraction algorithm, we propose a new generalisation of it.

In fact both Voronoy's master's thesis and his doctoral thesis were of such high quality that they were awarded the Bunyakovsky prize by the St Petersburg Academy of Sciences.

Later Voronoy worked on the theory of numbers, in particular he worked on algebraic numbers and the geometry of numbers. He extended work by Zolotarev and his work was the starting point for Vinogradov's investigations. His methods were also used by Hardy and Littlewood. His major works were Sur un problème du calcul des fonctions asympototiques Ⓣ (1903), Sur une fonction transcendente et ses applications à la sommation de quelques séries Ⓣ (1903), Sur quelques propriétés des formes quadratiques positives parfaites Ⓣ (1907), and Rescherches sur les paralleloèdres primitifs Ⓣ (1908). In the first of these he considered the sum from 1 to nn of t(i)t(i), the number of divisors of ii. This is equal to the number of integral points (x, y)(x,y) under the hyperbola xy = nxy=n.

Delone writes in [2] about Voronoy's work of 1907-08:

- Voronoy's memoir on parallelohedra represents one of the deepest investigations in the geometry of numbers in the world's literature, and the originality of the methods employed in the purely geometrical first part stamps the memoir with the imprint of genius.

In considering tessellations, Voronoy introduced what are today called Voronoi diagrams or Voronoi tessellations. Today they have wide applications to the analysis of spatially distributed data, so have become important in topics such as geophysics and meteorology. Although known under different names, the notion occurs in condensed matter physics, and in the study of Lie groups.

In 1904 Voronoy attended the Third International Congress of Mathematicians at Heidelberg. There he met Minkowski and they discovered that they were each working on similar topics.

His health deteriorated due to gallstones. He wrote in his diary:

- I am making great progress with the question under study [indefinite quadratic forms]; however, at the same time my health is becoming worse and worse. Yesterday I had for the first time a clear idea of the algorithm in the theory of forms I am investigating, but also suffered a strong attack of bilious colic which prevented me from working in the evening and from sleeping the whole night. I am so afraid that the results of my enduring efforts, obtained with such difficulty, will perish along with me.

Sadly his fears were justified and although he is known to have written a 28 page paper consisting of his notes on indefinite quadratic forms, this has been lost.


  1. I G Bashmakova, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
  2. B N Delone, The St Petersburg School of the Theory of Numbers (Moscow-Leningrad, 1947).
  3. E A Bolshakova and S S Ryshkov, On the surprise discovery of a manuscript of G F Voronoi (1928-1993) (Russian), Chebyshevskii Sb. 4 3(7) (2003), 171-181.
  4. A I Borodin, Mathematical calendar for the 1987/88 academic year (Russian), Mat. v Shkole (1) (1988), 82.
  5. Z I Krapivin, Georgii Feodosevich Voronoi (1868-1908) (Russian), Mat. v Shkole (1) (1989), 145-146.
  6. B V Pyaskovskii, The worldview of G F Voronoi and the methodology of his works (Russian), Ocherki Istor. Estestvoznan. Tekhn. No. 38 (1990), 38-43.
  7. G M Sita, The scientific legacy of Georgii Voronoi and contemporary science (Ukrainian), in Mathematics and the mathematical sciences in Ukraine in the twentieth century (Natsional. Akad. Nauk Ukraïni Inst. Mat., Kiev, 2001), 89-94.
  8. I Sh Slavutskii, Outline of the history of research on the arithmetic properties of Bernoulli numbers (von Staudt, Kummer, Voronoi) (Russian), Istor.-Mat. Issled. No. 32-33 (1990), 158-181.